{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:VRD6WJH3GHFS7NMO66UL4X33A6","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"1cf72db679cb0a8c3f78a45f8f0405ae22911390933db13030c5454f0eff0e77","cross_cats_sorted":["gr-qc","math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2011-07-25T20:00:02Z","title_canon_sha256":"d20149ee6dac179432d5b5b94d0802adc4cb971992677a971b18ebf9389eacc8"},"schema_version":"1.0","source":{"id":"1107.5041","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.5041","created_at":"2026-05-18T04:04:18Z"},{"alias_kind":"arxiv_version","alias_value":"1107.5041v4","created_at":"2026-05-18T04:04:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.5041","created_at":"2026-05-18T04:04:18Z"},{"alias_kind":"pith_short_12","alias_value":"VRD6WJH3GHFS","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"VRD6WJH3GHFS7NMO","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"VRD6WJH3","created_at":"2026-05-18T12:26:44Z"}],"graph_snapshots":[{"event_id":"sha256:1d64a73e700c286543e90105d06577bd43b5454163dd0a46a40a771e9e1fdead","target":"graph","created_at":"2026-05-18T04:04:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We construct a theory of fields living on continuous geometries with fractional Hausdorff and spectral dimensions, focussing on a flat background analogous to Minkowski spacetime. After reviewing the properties of fractional spaces with fixed dimension, presented in a companion paper, we generalize to a multi-fractional scenario inspired by multi-fractal geometry, where the dimension changes with the scale. This is related to the renormalization group properties of fractional field theories, illustrated by the example of a scalar field. Depending on the symmetries of the Lagrangian, one can de","authors_text":"Gianluca Calcagni","cross_cats":["gr-qc","math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2011-07-25T20:00:02Z","title":"Geometry and field theory in multi-fractional spacetime"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.5041","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f2633918f8c4ecb21a5063ed7a847816ee772b2f06e8cf11d4918311f89f420b","target":"record","created_at":"2026-05-18T04:04:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"1cf72db679cb0a8c3f78a45f8f0405ae22911390933db13030c5454f0eff0e77","cross_cats_sorted":["gr-qc","math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2011-07-25T20:00:02Z","title_canon_sha256":"d20149ee6dac179432d5b5b94d0802adc4cb971992677a971b18ebf9389eacc8"},"schema_version":"1.0","source":{"id":"1107.5041","kind":"arxiv","version":4}},"canonical_sha256":"ac47eb24fb31cb2fb58ef7a8be5f7b07a7d64214ffe4eec5ba090aa2d03f770c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ac47eb24fb31cb2fb58ef7a8be5f7b07a7d64214ffe4eec5ba090aa2d03f770c","first_computed_at":"2026-05-18T04:04:18.934268Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:04:18.934268Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VBuoLU85ItXNW5JR8iIwLfHdNgFEUpd/OS2SnjrFb3pMOYSra+CS7/l/uJNLP+7R6ihmUh7cQ+V7bYTh9590CA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:04:18.934869Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.5041","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f2633918f8c4ecb21a5063ed7a847816ee772b2f06e8cf11d4918311f89f420b","sha256:1d64a73e700c286543e90105d06577bd43b5454163dd0a46a40a771e9e1fdead"],"state_sha256":"fcdc9b3703d7b42796e390a80f20332fd47a4ec3c49f6856b86e65d6a5783ca1"}